If I cast a permanent again for 2 after airbending it, will its cast triggers be triggered again. Can I, for example, use [[Appa, Loyal Sky Bison]] to airbend [[Emrakul, the Promised End]], and take control of another player for the 2 recast cost? Or will even Emrakul’s cast cost reduction apply, allowing me to do this for even less?


Airbending allows the owner to “cast” from exile, so any effects that would happen upon casting will still happen.
Some effects specify “when cast from hand” or similar, and those won’t work, but the Emrakul in question should work as advertised.